Rigidity, Generators and Homology of Interval Exchange Groups

Abstract

Let $\Gamma$ be a dense countable subgroup of $\mathbb{R}$. Then, consider $IE(\Gamma )$; the group of piecewise linear bijections of $[0,1]$ with finitely many angles, all in $\Gamma$. We introduce and systematically study a family of partial transformation groupoids coming from inverse semigroups, ${\mathcal{G}}_{\Gamma }$, that realise $IE(\Gamma )$ as a topological full group. This new perspective on the groupoid models ${\mathcal{G}}_{\Gamma }$ of $IE(\Gamma )$ allows us to better understand the underlying C*-algebras and to compute homology. We show that $H_{\ast }({\mathcal{G}}_{\Gamma })=H_{\ast +1}(\Gamma )$. We show $C_r^{\ast }({\mathcal{G}}_{\Gamma })$ is classifiable in the sense of the Elliott classification program of C${}^{\ast }$-algebras. We then classify these groups via the Elliott invariant, showing $IE(\Gamma )\cong IE({\Gamma }^{\prime })\iff \Gamma ={\Gamma }^{\prime }$ as subsets of $\mathbb{R}$. We relate the K-Theory of the reduced C${}^{\ast }$ -algebras to groupoid homology via Matui's HK Conjecture. We relate the homology of $IE(\Gamma )$ to the homology of $\Gamma$ using the recent framework developed by Li. We investigate in greater detail three key cases, namely if $\Gamma \subset \mathbb{Q}$, if $\Gamma \cong {\mathbb{Z}}^n$, and if $\Gamma$ is a ring. For these three cases, we study homology in greater detail and find explicit generating sets.

26 Pages, comments welcome. Edit for minor corrections